Finding Power Series Representation for arctan(x): Help Needed!

AI Thread Summary
To find the power series representation for arctan(x), start with the integral of 1/(1 + x^2). Differentiating this function multiple times and evaluating at x = 0 can help derive the series. The discussion emphasizes the importance of focusing on the right-hand side to obtain the series for the left-hand side. It is suggested that one can integrate term by term after obtaining the power series for 1/(1 + x^2). Understanding these steps is crucial for successfully finding the power series representation.
teng125
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may i know how to solve this ques:find the power series representation for arctan (x)

i know that arctan (x) = integ 1/(1 + x^2) but then from here i don't know how to continue.
pls help...
 
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It seems to me if you're looking for the power series of the thing on the left hand side, then you might want to try looking for the power series of the thing on the right hand side.
 
ya I'm looking for the thing on the left hand side...pls help
 
Take the thing on the right, and differentiate it a few times, and let x = 0 each time.

f(x) = f(0) + xf'(0) + \frac{x^2}{2!}f''(0) + ...

Where f'(0) represents the derivative at x = 0. f''(0) is the second derivative etc. This will give a power series, then you can integrate term by term for the inverse tan function.
 
One doesn't need to know any calculus at all to find the power series for 1/(1+x^2).


ya I'm looking for the thing on the left hand side...pls help
I gave you a big hint -- have you not tried to do anything with it?
 
teng125 said:
ya I'm looking for the thing on the left hand side...pls help

Go back and read Hurkyl's reply again!
Hurkyl said:
It seems to me if you're looking for the power series of the thing on the left hand side, then you might want to try looking for the power series of the thing on the right hand side.
 

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